### Cartesian Double Categories

#### Posted by Mike Shulman

In general, there are two kinds of bicategories: those like $Cat$ and those like $Span$. In the $Cat$-like ones, the morphisms are “categorified functions”, which generally means some kind of “functor” between some kind of “category”, consisting of functions mapping objects and arrows from domain to codomain. But in the $Span$-like ones (which includes $Mod$ and $Prof$), the morphisms are not “functors” but rather some kind of “generalized relations” (including spans, modules, profunctors, and so on) which do not *map* from domain to codomain but rather *relate* the domain and codomain in some way.

In $Span$-like bicategories there is usually a subclass of the morphisms that *do* behave like categorified functions, and these play an important role. Usually the morphisms in this subclass all have right adjoints; sometimes they are exactly the morphisms with right adjoints; and often one can get away with talking about “morphisms with right adjoints” rather than making this subclass explicit. However, it’s also often conceptually and technically helpful to give the subclass as extra data, and arguably the most perspicuous way to do this is to work with a *double category* instead. This was the point of my first published paper, though others had certainly made the same point before, and I think more and more people are coming to recognize it.

Today a new installment in this story appeared on the arXiv: Cartesian Double Categories with an Emphasis on Characterizing Spans, by Evangelia Aleiferi. This is a project that I’ve wished for a while someone would do, so I’m excited that at last someone has!

We know now that various structure on a double category corresponds to similar structure on a bicategory. For instance, a monoidal structure on a (suitably well-behaved) double category induces a monoidal structure on its underlying bicategory. However, the monoidal double category is generally much stricter and easier to work with.

Aleiferi’s paper is about extending this to the *cartesian* monoidal case. A cartesian monoidal double category is easy to define: its diagonal $D\to D\times D$ and projection $D\to 1$ have right adjoints, just as for ordinary categories. It’s also easy to say what it means for a $Cat$-like bicategory to be cartesian monoidal: we can say that its diagonal and projection have right adjoints too, although that’s more complicated because the adjoints are generally only pseudofunctors living in a tricategory.

But it’s *not* at all obvious what it means for a $Span$-like bicategory to be “cartesian monoidal”. Intuitively, bicategories like $Span$ itself, or more generally $Span(E)$ for $E$ a category with finite limits, and $Prof(V)$ when $V$ is cartesian monoidal, should be “cartesian” — but they are not cartesian monoidal in the $Cat$-like way. The notion of cartesian bicategory was defined (by Carboni, Walters, Kelly, Verity, and Wood) to capture examples like these, but it is quite complicated. Moreover, to someone familiar with double categories, it is crying out to be reformulated in double-category language (e.g. it requires certain morphisms to have right adjoints, and induces $Cat$-like cartesian structure on the sub-bicategory of morphisms with right adjoints). In fact, it blows my mind that anyone was able to define the notion of cartesian bicategory without secretly having double categories in their head!

Aleiferi has now made a more careful study of cartesian *double* categories, and shown that they can be used for at least some (which I suspect will eventually become “nearly all”) of the same purposes as cartesian bicategories. For instance, here is a theorem from Lack-Walters-Wood Bicategories of spans as cartesian bicategories:

**Theorem:** A bicategory is equivalent to $Span(E)$, for some category $E$ with finite limits, if and only if it is cartesian, each comonad has an Eilenberg-Moore object, and every map is comonadic.

And here is a theorem from Aleiferi’s paper:

**Theorem:** A double category is equivalent to $Span(E)$, for some category $E$ with finite limits, if and only if it is cartesian, fibrant, unit-pure, and has strong Eilenberg-Moore objects for copointed endomorphisms.

Even without understanding all the words, the family resemblance should be clear, even if the technicalities are different. On a quick skim of Aleiferi’s paper it looks like there is no formal comparison yet between cartesian double categories and cartesian bicategories, but I’m sure that will come.

## Re: Cartesian double categories

Coincidently I’ve just heard Jens Seeber on cartesian bicategories used for Graphical conjunctive queries, which appear in database theory. Could cartesian double categories play a role here?